Commit 22633043 authored by niklas.baumgarten's avatar niklas.baumgarten
Browse files

added quivers

parent b4c6d588
Pipeline #145604 passed with stages
in 57 minutes and 58 seconds
%% Cell type:markdown id: tags:
# Elliptic Experiments
%% Cell type:markdown id: tags:
All results published in:
[1] Baumgarten, N. and Wieners, C. The parallel finite element system M++ with integrated multilevel
preconditioning and multilevel Monte Carlo methods. Computers & Mathematics with Applications.
2020
Main sources of this work:
[2] Giles, M.B., 2015. Multilevel Monte Carlo methods. Acta Numerica 24, 259–328.
[3] Dietrich, C.R., Newsam, G.N., 1997. Fast and exact simulation of stationary
Gaussian processes through circulant embedding of the covariance matrix.
SIAM Journal on Scientific Computing 18, 1088–1107.
[4] Charrier, J., Scheichl, R., Teckentrup, A.L., 2013. Finite element error analysis
of elliptic PDEs with random coefficients and its application to multilevel
Monte Carlo methods. SIAM Journal on Numerical Analysis 51, 322–352.
[5] Cliffe, K.A., Giles, M.B., Scheichl, R., Teckentrup, A.L., 2011. Multilevel Monte
Carlo methods and applications to elliptic PDEs with random coefficients.
Computing and Visualization in Science 14, 3.
%% Cell type:markdown id: tags:
### Stochastic subsurface flow problem
%% Cell type:markdown id: tags:
We apply the MLMC method an elliptic model problem in the case of a stochastically modeled permeability
$\kappa \colon \Omega \times \overline{D} \longrightarrow \mathbb{R}_{\text{sym}}^{d \times d}$
\begin{align}
\text{div} \, \big( \kappa(\omega, x_1, x_2) u(\omega, x_1, x_2) \big) &= 0, \quad \text{on} \quad D = (0,1)^2\\
u(\omega, x_1,x_2) &= 0, \quad \text{on} \quad x_2 = 0 \\
\nabla u(\omega, x_1,x_2) \cdot n &=-1, \quad \text{on} \quad x_2 = 1 \\
\nabla u(\omega, x_1,x_2) \cdot n &= 0, \quad \text{on} \quad x_1 \in \{ 0, 1 \}
\end{align}
on a suitable probability space $(\Omega, \mathcal{F}_\Omega , \mathbb{P})$.
We assume that the permeability tensor is isotropic and only depending on the scalar value
$\kappa_0(\omega, x)>0$. We choose a log-normal ansatz
\begin{align} \label{kappa_g}
\kappa_0(\omega, x) = \exp\big(g(\omega, x)\big)\,,
\end{align}
where $g \colon \Omega \times \bar D \rightarrow \mathbb{R}$ is a Gaussian field with a
covariance kernel characterized by
\begin{align} \label{covariance_fct}
C(x,y) = \sigma^2 \exp(- \| x- y \|_2^s / \lambda^s)
\end{align}
depending on variance $\sigma^2$, correlation length $\lambda$, and a smoothing parameter $s$.
For efficient sampling of these log-normal fields, we use the special structure of covariance matrices
with circulant embeddings and fast Fourier transforms.
The stochastic fields as above yield uniform regularity estimates for the elliptic model problem, thus for some $0 < t \leq 1$,
it is shown that
\begin{align}\label{regularity}
\| u(\omega, \cdot) &\|_{H^{r + 1}(D)} \lesssim \frac{\kappa_{\text{max}}(\omega) \| \kappa(\omega, \cdot) \|_{C^t(\overline{D})}^2}{\kappa_{\text{min}(\omega)^4}} \nonumber \\
&\left( \| f(\omega, \cdot) \|_{H^{t-1}(D)} + \| \kappa(\omega, \cdot) \|_{C^t(\overline{D})} \| u_{D}(\omega, \cdot) \|_{H^{t+\frac{1}{2}}(\Gamma_{D})} \right)
\end{align}
for almost all $\omega \in \Omega$ and for all $0 < r < t$. Therefore,
we can also retrieve convergence results for the finite element
solution $u_h(\omega)\in V_h^c(u_D)$.
%% Cell type:code id: tags:
``` python
import sys
sys.path.append('..')
import mpp.python.mppy as mppy
import mpp.python.mppy.utilities as utils
from python.mlmc_mppy import *
import pandas as pd
mpp = mppy.Mpp(project_name='MLMC',
executable='MLMC-M++',
kernels=4,
mute=False)
mpp.build()
```
%% Output
================ running cmake ================
-- Project directory= /home/niklas/CLionProjects/mlmc
-- Mpp directory= /home/niklas/CLionProjects/mlmc/mpp
-- Setting prepare commit message hook to mpp
file: /home/niklas/CLionProjects/mlmc/mpp/doc/../../.git/modules/mpp/hooks/prepare-commit-msg
-- Failed to find LLVM FileCheck
-- git Version: v1.5.2-bf585a27
-- Version: 1.5.2
-- Performing Test HAVE_STD_REGEX -- success
-- Performing Test HAVE_GNU_POSIX_REGEX -- failed to compile
-- Performing Test HAVE_POSIX_REGEX -- success
-- Performing Test HAVE_STEADY_CLOCK -- success
-- A library with LAPACK API found.
-- C Compiler optimization= -O3
-- C++ version= 17
-- Compiler version= c++
-- Compiler optimization= -O0
-- Suppress deprecated warnings= OFF
-- Building space time library= OFF
-- Time dimension= 0
-- Using fftw library= ON
-- Space dimension= 2
-- Affine linear transformations= ON
-- Geometric tolerance= 1e-10
-- Near zero= 1e-15
-- Very large= 1e30
-- Infinity= 1e100
-- Building mpp unit tests= OFF
-- Building mpp interval arithmetic unit tests= OFF
-- Configuring done
-- Generating done
-- Build files have been written to: /home/niklas/CLionProjects/mlmc/build
================ running make ================
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[ 99%] Built target TestMainMPI
Scanning dependencies of target TestSPRNG5MPI
[100%] Building CXX object mlmc/tests/CMakeFiles/TestSPRNG5MPI.dir/generators/TestSPRNG5.cpp.o
[100%] Linking CXX executable TestEmpiricMeasureLevelMaps
[100%] Linking CXX executable TestSPRNG5MPI
[100%] Built target TestSPRNG5MPI
[100%] Linking CXX executable TestExponents
[100%] Built target TestEmpiricMeasureLevelMaps
[100%] Linking CXX executable TestErrors
[100%] Built target TestExponents
[100%] Built target TestErrors
0
%% Cell type:markdown id: tags:
### Introductry example
%% Cell type:markdown id: tags:
As introductury example we want to take a look at the specific configuration $\sigma=1.0$, $\lambda=0.15$ and $s=1.8$. We solve this problem with linear Lagrange finite elements and initialize $l_{\text{init}} = [3, 4, 5]$ with $M_l^{\text{init}} = [12, 6, 3]$. We aim for an $\varepsilon = 0.01$ w.r.t the $L^2$-norm. First we are going to investigate the corresponding 1D problem before we will go ahead with the problem discribed above.
%% Cell type:markdown id: tags:
#### 1D
%% Cell type:code id: tags:
``` python
mpp.reset_data()
mpp.clean_data()
kwargs={"GeneratorPlotting": 1,
"PDESolverPlotting": 1,
"Problem": "StochasticLaplace1D",
"initLevels": [5, 6, 7]}
mpp.run(4, config="mlmc_elliptic", kwargs=kwargs)
```
%% Output
================ running mpp ================
start program on 4 procs at Tue Apr 13 09:49:05 2021
Running on: niklas-ThinkPad-T470p
Config Info:
GeneratorVerbose: ......... 0
NewtonVerbose: ............ 0
PDESolverVerbose: ......... 0
LinearVerbose: ............ 0
MeshVerbose: .............. 1
MainVerbose: .............. 1
MCVerbose: ................ 1
MLMCVerbose: .............. 1
PDESolverPlotting: ........ 1
GeneratorPlotting: ........ 1
lambda: ................... [0.15, 0.15]
Model: .................... LagrangeElliptic
AssembleVerbose: .......... 0
degree: ................... 1
MCParallel: ............... false
maxLevel: ................. 7
Problem: .................. StochasticLaplace1D
StochasticField: .......... LogNormal
epsilon: .................. 0.01
ConfigVerbose: ............ 1
onlyFine: ................. false
Mean: ..................... 0.0
initLevels: ............... [5, 6, 7]
smoothing: ................ 1.0
Quantity: ................. L2
initSampleAmount: ......... [12, 6, 3]
sigma: .................... 1.0
MLMC Experiment: Run
MLMC Method: eps=0.010000
MonteCarlo l=5: Start with: M=0 dM=12 Mcomm=0 dMcomm=12
MonteCarlo l=5: MeanQ=0 MeanY=0 SVarQ=0 SVarY=0
MonteCarlo l=5: End with: M=12 dM=12 Mcomm=0 dMcomm=0
MonteCarlo l=5: MeanQ=0.55705806 MeanY=0.55705806 SVarQ=0.009253913 SVarY=0.009253913
MonteCarlo l=5: Took 0.06 seconds
MonteCarlo l=6: Start with: M=0 dM=6 Mcomm=0 dMcomm=6
MonteCarlo l=6: MeanQ=0 MeanY=0 SVarQ=0 SVarY=0
MonteCarlo l=6: End with: M=6 dM=6 Mcomm=0 dMcomm=0
MonteCarlo l=6: MeanQ=0.57577254 MeanY=0.022917194 SVarQ=0.0049663677 SVarY=0.00082827073
MonteCarlo l=6: Took 0.07 seconds
MonteCarlo l=7: Start with: M=0 dM=3 Mcomm=0 dMcomm=3
MonteCarlo l=7: MeanQ=0 MeanY=0 SVarQ=0 SVarY=0
MonteCarlo l=7: End with: M=3 dM=3 Mcomm=0 dMcomm=0
MonteCarlo l=7: MeanQ=0.62666845 MeanY=0.020166238 SVarQ=0.0069772841 SVarY=0.0012463672
MonteCarlo l=7: Took 0.05 seconds
MLMC Method: alpha=0.1844884 beta=-0.58955487 gamma=0.95693128
MLMC Method: dM=[380, 80, 72]
MonteCarlo l=5: Start with: M=12 dM=380 Mcomm=0 dMcomm=380
MonteCarlo l=5: MeanQ=0.55705806 MeanY=0.55705806 SVarQ=0.009253913 SVarY=0.009253913
MonteCarlo l=5: End with: M=392 dM=380 Mcomm=0 dMcomm=0
MonteCarlo l=5: MeanQ=0.58077947 MeanY=0.58077947 SVarQ=0.0072659961 SVarY=0.0072659961
MonteCarlo l=5: Took 1.48 seconds
MonteCarlo l=6: Start with: M=6 dM=80 Mcomm=0 dMcomm=80
MonteCarlo l=6: MeanQ=0.57577254 MeanY=0.022917194 SVarQ=0.0049663677 SVarY=0.00082827073
MonteCarlo l=6: End with: M=86 dM=80 Mcomm=0 dMcomm=0
MonteCarlo l=6: MeanQ=0.58638908 MeanY=0.0020076219 SVarQ=0.0082427325 SVarY=0.0028916966
MonteCarlo l=6: Took 0.79 seconds
MonteCarlo l=7: Start with: M=3 dM=72 Mcomm=0 dMcomm=72
MonteCarlo l=7: MeanQ=0.62666845 MeanY=0.020166238 SVarQ=0.0069772841 SVarY=0.0012463672
MonteCarlo l=7: End with: M=75 dM=72 Mcomm=0 dMcomm=0
MonteCarlo l=7: MeanQ=0.5817257 MeanY=0.0029040982 SVarQ=0.0054958335 SVarY=0.0017595161
MonteCarlo l=7: Took 1.24 seconds
MLMC Method: alpha=-0.53260264 beta=0.71673745 gamma=0.95693128
MLMC Method: dM=[17, 102, 30]
MonteCarlo l=5: Start with: M=392 dM=17 Mcomm=0 dMcomm=17
MonteCarlo l=5: MeanQ=0.58077947 MeanY=0.58077947 SVarQ=0.0072659961 SVarY=0.0072659961
MonteCarlo l=5: End with: M=409 dM=17 Mcomm=0 dMcomm=0
MonteCarlo l=5: MeanQ=0.58047082 MeanY=0.58047082 SVarQ=0.0073352332 SVarY=0.0073352332
MonteCarlo l=5: Took 0.07 seconds
MonteCarlo l=6: Start with: M=86 dM=102 Mcomm=0 dMcomm=102
MonteCarlo l=6: MeanQ=0.58638908 MeanY=0.0020076219 SVarQ=0.0082427325 SVarY=0.0028916966
MonteCarlo l=6: End with: M=188 dM=102 Mcomm=0 dMcomm=0
MonteCarlo l=6: MeanQ=0.58086857 MeanY=0.0044612122 SVarQ=0.0073905344 SVarY=0.0029985566
MonteCarlo l=6: Took 1.03 seconds
MonteCarlo l=7: Start with: M=75 dM=30 Mcomm=0 dMcomm=30
MonteCarlo l=7: MeanQ=0.5817257 MeanY=0.0029040982 SVarQ=0.0054958335 SVarY=0.0017595161
MonteCarlo l=7: End with: M=105 dM=30 Mcomm=0 dMcomm=0
MonteCarlo l=7: MeanQ=0.5867427 MeanY=0.00099310045 SVarQ=0.0050168801 SVarY=0.0018706446
MonteCarlo l=7: Took 0.53 seconds
MLMC Method: alpha=2.1674242 beta=0.68073273 gamma=0.95693128
MLMC Method: dM=[9, 7, 6]
MonteCarlo l=5: Start with: M=409 dM=9 Mcomm=0 dMcomm=9
MonteCarlo l=5: MeanQ=0.58047082 MeanY=0.58047082 SVarQ=0.0073352332 SVarY=0.0073352332
MonteCarlo l=5: End with: M=418 dM=9 Mcomm=0 dMcomm=0
MonteCarlo l=5: MeanQ=0.58122909 MeanY=0.58122909 SVarQ=0.0072965256 SVarY=0.0072965256
MonteCarlo l=5: Took 0.04 seconds
MonteCarlo l=6: Start with: M=188 dM=7 Mcomm=0 dMcomm=7
MonteCarlo l=6: MeanQ=0.58086857 MeanY=0.0044612122 SVarQ=0.0073905344 SVarY=0.0029985566
MonteCarlo l=6: End with: M=195 dM=7 Mcomm=0 dMcomm=0
MonteCarlo l=6: MeanQ=0.58150053 MeanY=0.0042596788 SVarQ=0.0075483522 SVarY=0.0029239404
MonteCarlo l=6: Took 0.08 seconds
MonteCarlo l=7: Start with: M=105 dM=6 Mcomm=0 dMcomm=6
MonteCarlo l=7: MeanQ=0.5867427 MeanY=0.00099310045 SVarQ=0.0050168801 SVarY=0.0018706446
MonteCarlo l=7: End with: M=111 dM=6 Mcomm=0 dMcomm=0
MonteCarlo l=7: MeanQ=0.58340208 MeanY=0.001693584 SVarQ=0.0051377899 SVarY=0.0018255814
MonteCarlo l=7: Took 0.11 seconds
MLMC Method: alpha=1.3306651 beta=0.67955787 gamma=0.95693128
MLMC Method: dM=[-5, -4, -3]
MLMC Method: numErr=0.001405662
MLMC Method: statErr=0.0069926425
MLMC Method: 5.49 seconds
MLMC Experiment: 5.49 seconds
MC Results Info:
E(Qf-Qc): ................. [0.581229, 0.00425968, 0.00169358]
E(Qf): .................... [0.581229, 0.581501, 0.583402]
V(Qf-Qc): ................. [0.00729653, 0.00292394, 0.00182558]
V(Qf): .................... [0.00729653, 0.00754835, 0.00513779]
kurtosis: ................. [2.56877, 3.99293, 2.8725]
E(cost): .................. [9, 17, 33]
Used Levels: .............. [5, 6, 7]
Used Samples: ............. [418, 195, 111]
Exponents Info:
Final alpha: .............. 1.3306651
Final beta: ............... 0.67955787
Final gamma: .............. 0.95693128
MLMC Results Info:
Value: .................... 2.9461225
Cost: ..................... 295
Epsilon: .................. 0.01
Total Error: .............. 0.0083983045
Statistical Error: ........ 0.0069926425
Numerical Error: .......... 0.001405662
end program after 5.57 seconds on 4 procs at Tue Apr 13 09:49:11 2021
0
%% Cell type:code id: tags:
``` python
mpp.reset_data()
mpp.parse_log()
convergence_plot(mpp)
mlmc_plot(mpp)
```
%% Output
%% Cell type:code id: tags:
``` python
import os
len(os.listdir(mpp.dm.PROJECT_VTU_DATA_DIR))
```
%% Output
3090
%% Cell type:code id: tags:
``` python
from mpp.python.mppy.utilities import VtkPlot
from ipywidgets import interact, interactive, fixed, interact_manual
import ipywidgets as widgets
@interact(n=widgets.IntSlider(min=0, max=5, step=1, value=0))
def plot_nth_frame(n):
p = VtkPlot(figsize=(16, 6))
plt.axis('off')
gs = gridspec.GridSpec(1, 2)
p.ax1 = p.fig.add_subplot(gs[0])
p.add_1d_vtu("Kappa.6.0.{}.vtu".format(n), ax=p.ax1)
p.ax2 = p.fig.add_subplot(gs[1])
p.add_1d_vtu("Kappa.6.1.{}.vtu".format(n), ax=p.ax2)
```
%% Output
%% Cell type:code id: tags:
``` python
from mpp.python.mppy.utilities import VtkPlot
from ipywidgets import interact, interactive, fixed, interact_manual
import ipywidgets as widgets
@interact(n=widgets.IntSlider(min=0, max=5, step=1, value=0))
def plot_nth_frame(n):
p = VtkPlot(figsize=(16, 6))
plt.axis('off')
gs = gridspec.GridSpec(1, 2)
p.ax1 = p.fig.add_subplot(gs[0])
p.add_1d_vtu("Kappa.7.0.{}.vtu".format(n), ax=p.ax1)
p.ax2 = p.fig.add_subplot(gs[1])
p.add_1d_vtu("Kappa.7.1.{}.vtu".format(n), ax=p.ax2)
```
%% Output
%% Cell type:markdown id: tags:
#### 2D
%% Cell type:code id: tags:
``` python
mpp.mute = False
mpp.build()
mpp.mute = False
mpp.reset_data()
mpp.clean_data()
kwargs={"GeneratorPlotting": 1,
"PDESolverPlotting": 1,
"initLevels": [4, 5, 6]}
mpp.run(4, config="mlmc_elliptic", kwargs=kwargs)
```
%% Output
================ running cmake ================
-- Project directory= /home/niklas/CLionProjects/mlmc
-- Mpp directory= /home/niklas/CLionProjects/mlmc/mpp
-- Setting prepare commit message hook to mpp
file: /home/niklas/CLionProjects/mlmc/mpp/doc/../../.git/modules/mpp/hooks/prepare-commit-msg
-- Failed to find LLVM FileCheck
-- git Version: v1.5.2-bf585a27
-- Version: 1.5.2
-- Performing Test HAVE_STD_REGEX -- success
-- Performing Test HAVE_GNU_POSIX_REGEX -- failed to compile
-- Performing Test HAVE_POSIX_REGEX -- success
-- Performing Test HAVE_STEADY_CLOCK -- success
-- A library with LAPACK API found.
-- C Compiler optimization= -O3
-- C++ version= 17
-- Compiler version= c++
-- Compiler optimization= -O0
-- Suppress deprecated warnings= OFF
-- Building space time library= OFF
-- Time dimension= 0
-- Using fftw library= ON
-- Space dimension= 2
-- Affine linear transformations= ON
-- Geometric tolerance= 1e-10
-- Near zero= 1e-15
-- Very large= 1e30
-- Infinity= 1e100
-- Building mpp unit tests= OFF
-- Building mpp interval arithmetic unit tests= OFF
-- Configuring done
-- Generating done
-- Build files have been written to: /home/niklas/CLionProjects/mlmc/build
================ running make ================
[ 1%] Built target gtest
[ 6%] Built target benchmark
[ 10%] Built target LIB_BASIC
[ 57%] Built target superlu
[ 58%] Built target gmock
[ 58%] Built target benchmark_main
[ 58%] Built target gtest_main
[ 58%] Built target gmock_main
[ 62%] Built target LIB_MATH
[ 63%] Built target LIB_TEST
[ 68%] Built target LIB_MESH
[ 71%] Built target LIB_DISC
[ 75%] Built target LIB_VECTOR
[ 80%] Built target LIB_ELEM
[ 84%] Built target LIB_SOLVERS
[ 84%] Built target MPP_LIBRARIES
[ 84%] Built target BASICS
[ 84%] Linking CXX executable TestLevel
[ 84%] Linking CXX executable TestPlotting
[ 87%] Built target PDESOLVER
[ 88%] Built target PROBLEMS
[ 90%] Built target GENERATORS
[ 90%] Linking CXX executable TestNormalDistribution
[ 92%] Built target MONTECARLO
[ 92%] Linking CXX executable TestCirculantEmbedding
[ 92%] Built target TestLevel
[ 92%] Linking CXX executable TestSymmetricCovariance
[ 93%] Built target TestNormalDistribution
[ 94%] Built target TestMainMPI
[ 94%] Linking CXX executable TestEmpiricMeasureLevelMaps
[ 94%] Built target TestSymmetricCovariance
[ 94%] Linking CXX executable TestPDESolverMPI
[ 94%] Built target TestPlotting
[ 94%] Linking CXX executable TestExponents
[ 94%] Built target TestCirculantEmbedding
[ 94%] Linking CXX executable TestPDESolver
[ 95%] Built target TestEmpiricMeasureLevelMaps
[ 95%] Linking CXX executable TestErrors
[ 96%] Built target TestPDESolverMPI
[ 96%] Linking CXX executable TestSPRNG5MPI
[ 97%] Built target TestExponents
[ 97%] Built target TestPDESolver
[ 97%] Linking CXX executable MLMC-M++
[ 97%] Linking CXX executable TestWelfordAggregateMPI
[ 98%] Built target TestSPRNG5MPI
[ 98%] Linking CXX executable TestMonteCarloMPI
[ 99%] Built target TestWelfordAggregateMPI
[ 99%] Built target TestErrors
[100%] Built target MLMC-M++
[100%] Built target TestMonteCarloMPI
================ running mpp ================
start program on 4 procs at Tue Apr 13 10:16:20 2021
Running on: niklas-ThinkPad-T470p
Config Info:
GeneratorVerbose: ......... 0
NewtonVerbose: ............ 0
PDESolverVerbose: ......... 0
LinearVerbose: ............ 0
MeshVerbose: .............. 1
MainVerbose: .............. 1
MCVerbose: ................ 1
MLMCVerbose: .............. 1
PDESolverPlotting: ........ 1
GeneratorPlotting: ........ 1
lambda: ................... [0.15, 0.15]
Model: .................... LagrangeElliptic
AssembleVerbose: .......... 0
degree: ................... 1
MCParallel: ............... false
maxLevel: ................. 7
Problem: .................. StochasticLaplace2D
StochasticField: .......... LogNormal
epsilon: .................. 0.01
ConfigVerbose: ............ 1
onlyFine: ................. false
Mean: ..................... 0.0
initLevels: ............... [4, 5, 6]
smoothing: ................ 1.0
Quantity: ................. L2
initSampleAmount: ......... [12, 6, 3]
sigma: .................... 1.0
MLMC Experiment: Run
MLMC Method: eps=0.010000
MonteCarlo l=4: Start with: M=0 dM=12 Mcomm=0 dMcomm=12
MonteCarlo l=4: MeanQ=0 MeanY=0 SVarQ=0 SVarY=0
MonteCarlo l=4: End with: M=12 dM=12 Mcomm=0 dMcomm=0
MonteCarlo l=4: MeanQ=0.57403875 MeanY=0.57403875 SVarQ=0.00089497086 SVarY=0.00089497086
MonteCarlo l=4: Took 0.89 seconds
MonteCarlo l=5: Start with: M=0 dM=6 Mcomm=0 dMcomm=6
MonteCarlo l=5: MeanQ=0 MeanY=0 SVarQ=0 SVarY=0
MonteCarlo l=5: End with: M=6 dM=6 Mcomm=0 dMcomm=0
MonteCarlo l=5: MeanQ=0.57188341 MeanY=2.1976422e-05 SVarQ=0.00089218133 SVarY=3.2914846e-07
MonteCarlo l=5: Took 2.56 seconds
MonteCarlo l=6: Start with: M=0 dM=3 Mcomm=0 dMcomm=3
MonteCarlo l=6: MeanQ=0 MeanY=0 SVarQ=0 SVarY=0
Warning: Kurtosis of Y above 100!
in /home/niklas/CLionProjects/mlmc/mlmc/src/montecarlo/datastructure/EmpiricMeasures.cpp:42
MonteCarlo l=6: End with: M=3 dM=3 Mcomm=0 dMcomm=0
MonteCarlo l=6: MeanQ=0.5621161 MeanY=0.00018661905 SVarQ=0.00076408299 SVarY=1.2266638e-08
MonteCarlo l=6: Took 8.38 seconds
MLMC Method: alpha=-3.0860678 beta=4.7459266 gamma=1.9138626
MLMC Method: dM=[7, -4, -1]
MonteCarlo l=4: Start with: M=12 dM=7 Mcomm=0 dMcomm=7
MonteCarlo l=4: MeanQ=0.57403875 MeanY=0.57403875 SVarQ=0.00089497086 SVarY=0.00089497086
MonteCarlo l=4: End with: M=19 dM=7 Mcomm=0 dMcomm=0
MonteCarlo l=4: MeanQ=0.57197445 MeanY=0.57197445 SVarQ=0.0013728365 SVarY=0.0013728365
MonteCarlo l=4: Took 0.66 seconds
MLMC Method: alpha=-3.0860678 beta=4.7459266 gamma=1.9138626
MLMC Method: dM=[10, -4, -1]
MonteCarlo l=4: Start with: M=19 dM=10 Mcomm=0 dMcomm=10
MonteCarlo l=4: MeanQ=0.57197445 MeanY=0.57197445 SVarQ=0.0013728365 SVarY=0.0013728365
MonteCarlo l=4: End with: M=29 dM=10 Mcomm=0 dMcomm=0
MonteCarlo l=4: MeanQ=0.5744856 MeanY=0.5744856 SVarQ=0.0013184221 SVarY=0.0013184221
MonteCarlo l=4: Took 0.93 seconds
MLMC Method: alpha=-3.0860678 beta=4.7459266 gamma=1.9138626
MLMC Method: dM=[-1, -4, -1]
MLMC Method: numErr=0
MLMC Method: statErr=0.0067469829
MLMC Method: 13.40 seconds
MLMC Experiment: 13.40 seconds
MC Results Info:
E(Qf-Qc): ................. [0.574486, 2.19764e-05, 0.000186619]
E(Qf): .................... [0.574486, 0.571883, 0.562116]
V(Qf-Qc): ................. [0.00131842, 3.29148e-07, 1.22666e-08]
V(Qf): .................... [0.00131842, 0.000892181, 0.000764083]
kurtosis: ................. [3.00616, 2.17199, 138.007]
E(cost): .................. [81, 289, 1089]
Used Levels: .............. [4, 5, 6]
Used Samples: ............. [29, 6, 3]
Exponents Info:
Final alpha: .............. -3.0860678
Final beta: ............... 4.7459266
Final gamma: .............. 1.9138626
MLMC Results Info:
Value: .................... 2.2958188
Cost: ..................... 5836
Epsilon: .................. 0.01
Total Error: .............. 0.0067469829
Statistical Error: ........ 0.0067469829
Numerical Error: .......... 0
end program after 14.11 seconds on 4 procs at Tue Apr 13 10:16:35 2021
0
%% Cell type:code id: tags:
``` python
mpp.parse_log()
convergence_plot(mpp)
mlmc_plot(mpp)
```
%% Output
%% Cell type:code id: tags:
``` python
import os
len(os.listdir(mpp.dm.PROJECT_VTU_DATA_DIR))
```
%% Output
141
%% Cell type:code id: tags:
``` python
from mpp.python.mppy.utilities import VtkPlot
from ipywidgets import interact, interactive, fixed, interact_manual
import ipywidgets as widgets
@interact(n=widgets.IntSlider(min=0, max=5, step=1, value=0))
def plot_nth_frame(n):
p = VtkPlot(figsize=(16, 12))
plt.axis('off')
gs = gridspec.GridSpec(2, 2)
p.ax1 = p.fig.add_subplot(gs[0])
p.add_vtu("Kappa.5.0.{}.vtu".format(n), ax=p.ax1)
p.add_quivers("Flux.5.0.{}.vtu".format(n), ax=p.ax1)
p.ax2 = p.fig.add_subplot(gs[1])
p.add_vtu("Kappa.5.1.{}.vtu".format(n), ax=p.ax2)
p.add_quivers("Flux.5.1.{}.vtu".format(n), ax=p.ax2)
p.ax3 = p.fig.add_subplot(gs[2])
p.add_vtu("Pressure.5.0.{}.vtu".format(n), ax=p.ax3)
p.ax4 = p.fig.add_subplot(gs[3])
p.add_vtu("Pressure.5.1.{}.vtu".format(n), ax=p.ax4)
```
%% Output
%% Cell type:code id: tags:
``` python
from mpp.python.mppy.utilities import VtkPlot
from ipywidgets import interact, interactive, fixed, interact_manual
import ipywidgets as widgets
@interact(n=widgets.IntSlider(min=0, max=2, step=1, value=0))
def plot_nth_frame(n):
p = VtkPlot(figsize=(16, 12))
plt.axis('off')
gs = gridspec.GridSpec(2, 2)
p.ax1 = p.fig.add_subplot(gs[0])
p.add_vtu("Kappa.6.0.{}.vtu".format(n), ax=p.ax1)
p.add_quivers("Flux.6.0.{}.vtu".format(n), ax=p.ax1)
p.ax2 = p.fig.add_subplot(gs[1])
p.add_vtu("Kappa.6.1.{}.vtu".format(n), ax=p.ax2)
p.add_quivers("Flux.6.1.{}.vtu".format(n), ax=p.ax2)
p.ax3 = p.fig.add_subplot(gs[2])
p.add_vtu("Pressure.6.0.{}.vtu".format(n), ax=p.ax3)
p.ax4 = p.fig.add_subplot(gs[3])
p.add_vtu("Pressure.6.1.{}.vtu".format(n), ax=p.ax4)
```
%% Output
%% Cell type:markdown id: tags:
## Convergence tests
%% Cell type:code id: tags:
``` python
test_case = True
if test_case:
mpp.kernels = 4
mpp.mute = False
kwargs = {"epsilon": 0.0,
"initLevels": [3, 4, 5, 6, 7],
"initSampleAmount": [100, 100, 100, 100, 100]}
epsilons = [0.1, 0.03, 0.01, 0.003]
else:
mpp.kernels = 32
mpp.mute = True
kwargs = {"epsilon": 0.0,
"initLevels": [3, 4, 5, 6, 7, 8, 9],
"initSampleAmount": [500, 500, 500, 500, 500, 500, 500]}
epsilons = [0.1, 0.03, 0.01, 0.003, 0.001, 0.0003]
```
%% Cell type:markdown id: tags:
### Dependency on $s$
%% Cell type:code id: tags:
``` python
mpp.clean_data()
mpp.reset_data()
smoothings = [1.0, 1.4, 1.8]
for smoothing in smoothings:
kwargs["smoothing"] = smoothing
mpp.run(config="mlmc_elliptic", kwargs=kwargs)
mpp.parse_log()
convergence_plot(mpp)
```
%% Output
================ running mpp ================
start program on 4 procs at Thu Apr 8 17:15:16 2021
Running on: niklas-ThinkPad-T470p
EllipticPDESolver Info:
Solver Info:
Assemble Info:
Assemble Name: ............ LagrangeEllipticAssemble
Discretization: ........... P1
Problem: .................. StochasticLaplace2D
EllipticPDESolver Info:
Solver Info:
Assemble Info:
Assemble Name: ............ LagrangeEllipticAssemble
Discretization: ........... P1
Problem: .................. StochasticLaplace2D
EllipticPDESolver Info:
Solver Info:
Assemble Info:
Assemble Name: ............ LagrangeEllipticAssemble
Discretization: ........... P1
Problem: .................. StochasticLaplace2D
EllipticPDESolver Info:
Solver Info:
Assemble Info:
Assemble Name: ............ LagrangeEllipticAssemble
Discretization: ........... P1
Problem: .................. StochasticLaplace2D
Invalid MIT-MAGIC-COOKIE-1 key[niklas-ThinkPad-T470p:215298] *** Process received signal ***
[niklas-ThinkPad-T470p:215298] Signal: Segmentation fault (11)
[niklas-ThinkPad-T470p:215298] Signal code: Address not mapped (1)
[niklas-ThinkPad-T470p:215298] Failing at address: 0x55a00d921000
[niklas-ThinkPad-T470p:215298] [ 0] /lib/x86_64-linux-gnu/libpthread.so.0(+0x153c0)[0x7f62be56d3c0]
[niklas-ThinkPad-T470p:215298] [ 1] MLMC-M++(+0x11022c)[0x55a00b9ba22c]
[niklas-ThinkPad-T470p:215298] [ 2] MLMC-M++(+0x10ebd7)[0x55a00b9b8bd7]
[niklas-ThinkPad-T470p:215298] [ 3] MLMC-M++(+0x106c9d)[0x55a00b9b0c9d]
[niklas-ThinkPad-T470p:215298] [ 4] MLMC-M++(+0x104477)[0x55a00b9ae477]
[niklas-ThinkPad-T470p:215298] [ 5] MLMC-M++(+0xfc6e9)[0x55a00b9a66e9]
[niklas-ThinkPad-T470p:215298] [ 6] MLMC-M++(+0xf8547)[0x55a00b9a2547]
[niklas-ThinkPad-T470p:215298] [ 7] MLMC-M++(+0xeb08d)[0x55a00b99508d]
[niklas-ThinkPad-T470p:215298] [ 8] MLMC-M++(+0xeb162)[0x55a00b995162]
[niklas-ThinkPad-T470p:215298] [ 9] MLMC-M++(+0xf014e)[0x55a00b99a14e]
[niklas-ThinkPad-T470p:215298] [10] MLMC-M++(+0xf081c)[0x55a00b99a81c]
[niklas-ThinkPad-T470p:215298] [11] MLMC-M++(+0xee5d9)[0x55a00b9985d9]
[niklas-ThinkPad-T470p:215298] [12] MLMC-M++(+0x71e9c)[0x55a00b91be9c]
[niklas-ThinkPad-T470p:215298] [13] MLMC-M++(+0x53593)[0x55a00b8fd593]
[niklas-ThinkPad-T470p:215298] [14] MLMC-M++(+0x53322)[0x55a00b8fd322]
[niklas-ThinkPad-T470p:215298] [15] MLMC-M++(+0x4e3c8)[0x55a00b8f83c8]
[niklas-ThinkPad-T470p:215298] [16] MLMC-M++(+0x3e979)[0x55a00b8e8979]
[niklas-ThinkPad-T470p:215298] [17] MLMC-M++(+0x3ebca)[0x55a00b8e8bca]
[niklas-ThinkPad-T470p:215298] [18] MLMC-M++(+0x369a7)[0x55a00b8e09a7]
[niklas-ThinkPad-T470p:215298] [19] /lib/x86_64-linux-gnu/libc.so.6(__libc_start_main+0xf3)[0x7f62be38d0b3]
[niklas-ThinkPad-T470p:215298] [20] MLMC-M++(+0x34c6e)[0x55a00b8dec6e]
[niklas-ThinkPad-T470p:215298] *** End of error message ***
--------------------------------------------------------------------------
Primary job terminated normally, but 1 process returned
a non-zero exit code. Per user-direction, the job has been aborted.
--------------------------------------------------------------------------
--------------------------------------------------------------------------
mpirun noticed that process rank 0 with PID 0 on node niklas-ThinkPad-T470p exited on signal 11 (Segmentation fault).
--------------------------------------------------------------------------
================ running mpp ================
start program on 4 procs at Thu Apr 8 17:15:30 2021
Running on: niklas-ThinkPad-T470p
EllipticPDESolver Info:
Solver Info:
Assemble Info:
Assemble Name: ............ LagrangeEllipticAssemble
Discretization: ........... P1
Problem: .................. StochasticLaplace2D
EllipticPDESolver Info:
Solver Info:
Assemble Info:
Assemble Name: ............ LagrangeEllipticAssemble
Discretization: ........... P1
Problem: .................. StochasticLaplace2D
EllipticPDESolver Info:
Solver Info:
Assemble Info:
Assemble Name: ............ LagrangeEllipticAssemble
Discretization: ........... P1
Problem: .................. StochasticLaplace2D
EllipticPDESolver Info:
Solver Info:
Assemble Info:
Assemble Name: ............ LagrangeEllipticAssemble
Discretization: ........... P1
Problem: .................. StochasticLaplace2D
Invalid MIT-MAGIC-COOKIE-1 key[niklas-ThinkPad-T470p:215323] *** Process received signal ***
[niklas-ThinkPad-T470p:215323] Signal: Segmentation fault (11)
[niklas-ThinkPad-T470p:215323] Signal code: Address not mapped (1)
[niklas-ThinkPad-T470p:215323] Failing at address: 0x5646fb8ff000
[niklas-ThinkPad-T470p:215323] [ 0] /lib/x86_64-linux-gnu/libpthread.so.0(+0x153c0)[0x7f0de24b93c0]
[niklas-ThinkPad-T470p:215323] [ 1] MLMC-M++(+0x11022c)[0x5646f9c1d22c]
[niklas-ThinkPad-T470p:215323] [ 2] MLMC-M++(+0x10ebd7)[0x5646f9c1bbd7]
[niklas-ThinkPad-T470p:215323] [ 3] MLMC-M++(+0x106c9d)[0x5646f9c13c9d]
[niklas-ThinkPad-T470p:215323] [ 4] MLMC-M++(+0x104477)[0x5646f9c11477]
[niklas-ThinkPad-T470p:215323] [ 5] MLMC-M++(+0xfc6e9)[0x5646f9c096e9]
[niklas-ThinkPad-T470p:215323] [ 6] MLMC-M++(+0xf8547)[0x5646f9c05547]
[niklas-ThinkPad-T470p:215323] [ 7] MLMC-M++(+0xeb08d)[0x5646f9bf808d]
[niklas-ThinkPad-T470p:215323] [ 8] MLMC-M++(+0xeb162)[0x5646f9bf8162]
[niklas-ThinkPad-T470p:215323] [ 9] MLMC-M++(+0xf014e)[0x5646f9bfd14e]
[niklas-ThinkPad-T470p:215323] [10] MLMC-M++(+0xf081c)[0x5646f9bfd81c]
[niklas-ThinkPad-T470p:215323] [11] MLMC-M++(+0xee5d9)[0x5646f9bfb5d9]
[niklas-ThinkPad-T470p:215323] [12] MLMC-M++(+0x71e9c)[0x5646f9b7ee9c]
[niklas-ThinkPad-T470p:215323] [13] MLMC-M++(+0x53593)[0x5646f9b60593]
[niklas-ThinkPad-T470p:215323] [14] MLMC-M++(+0x53322)[0x5646f9b60322]
[niklas-ThinkPad-T470p:215323] [15] MLMC-M++(+0x4e3c8)[0x5646f9b5b3c8]
[niklas-ThinkPad-T470p:215323] [16] MLMC-M++(+0x3e979)[0x5646f9b4b979]
[niklas-ThinkPad-T470p:215323] [17] MLMC-M++(+0x3ebca)[0x5646f9b4bbca]
[niklas-ThinkPad-T470p:215323] [18] MLMC-M++(+0x369a7)[0x5646f9b439a7]
[niklas-ThinkPad-T470p:215323] [19] /lib/x86_64-linux-gnu/libc.so.6(__libc_start_main+0xf3)[0x7f0de22d90b3]
[niklas-ThinkPad-T470p:215323] [20] MLMC-M++(+0x34c6e)[0x5646f9b41c6e]
[niklas-ThinkPad-T470p:215323] *** End of error message ***
--------------------------------------------------------------------------
Primary job terminated normally, but 1 process returned
a non-zero exit code. Per user-direction, the job has been aborted.
--------------------------------------------------------------------------
--------------------------------------------------------------------------
mpirun noticed that process rank 0 with PID 0 on node niklas-ThinkPad-T470p exited on signal 11 (Segmentation fault).
--------------------------------------------------------------------------
================ running mpp ================
start program on 4 procs at Thu Apr 8 17:15:55 2021
Running on: niklas-ThinkPad-T470p
EllipticPDESolver Info:
Solver Info:
Assemble Info:
Assemble Name: ............ LagrangeEllipticAssemble
Discretization: ........... P1
Problem: .................. StochasticLaplace2D
EllipticPDESolver Info:
Solver Info:
Assemble Info:
Assemble Name: ............ LagrangeEllipticAssemble
Discretization: ........... P1
Problem: .................. StochasticLaplace2D
EllipticPDESolver Info:
Solver Info:
Assemble Info:
Assemble Name: ............ LagrangeEllipticAssemble
Discretization: ........... P1
Problem: .................. StochasticLaplace2D
EllipticPDESolver Info:
Solver Info:
Assemble Info:
Assemble Name: ............ LagrangeEllipticAssemble
Discretization: ........... P1
Problem: .................. StochasticLaplace2D
Invalid MIT-MAGIC-COOKIE-1 key[niklas-ThinkPad-T470p:215359] *** Process received signal ***
[niklas-ThinkPad-T470p:215359] Signal: Segmentation fault (11)
[niklas-ThinkPad-T470p:215359] Signal code: Address not mapped (1)
[niklas-ThinkPad-T470p:215359] Failing at address: 0x55f646b92000
[niklas-ThinkPad-T470p:215359] [ 0] /lib/x86_64-linux-gnu/libpthread.so.0(+0x153c0)[0x7f80bbfe83c0]
[niklas-ThinkPad-T470p:215359] [ 1] MLMC-M++(+0x11022c)[0x55f643f0922c]
[niklas-ThinkPad-T470p:215359] [ 2] MLMC-M++(+0x10ebd7)[0x55f643f07bd7]
[niklas-ThinkPad-T470p:215359] [ 3] MLMC-M++(+0x106c9d)[0x55f643effc9d]
[niklas-ThinkPad-T470p:215359] [ 4] MLMC-M++(+0x104477)[0x55f643efd477]
[niklas-ThinkPad-T470p:215359] [ 5] MLMC-M++(+0xfc6e9)[0x55f643ef56e9]
[niklas-ThinkPad-T470p:215359] [ 6] MLMC-M++(+0xf8547)[0x55f643ef1547]
[niklas-ThinkPad-T470p:215359] [ 7] MLMC-M++(+0xeb08d)[0x55f643ee408d]
[niklas-ThinkPad-T470p:215359] [ 8] MLMC-M++(+0xeb162)[0x55f643ee4162]
[niklas-ThinkPad-T470p:215359] [ 9] MLMC-M++(+0xf014e)[0x55f643ee914e]
[niklas-ThinkPad-T470p:215359] [10] MLMC-M++(+0xf081c)[0x55f643ee981c]
[niklas-ThinkPad-T470p:215359] [11] MLMC-M++(+0xee5d9)[0x55f643ee75d9]
[niklas-ThinkPad-T470p:215359] [12] MLMC-M++(+0x71e9c)[0x55f643e6ae9c]
[niklas-ThinkPad-T470p:215359] [13] MLMC-M++(+0x53593)[0x55f643e4c593]
[niklas-ThinkPad-T470p:215359] [14] MLMC-M++(+0x53322)[0x55f643e4c322]
[niklas-ThinkPad-T470p:215359] [15] MLMC-M++(+0x4e3c8)[0x55f643e473c8]
[niklas-ThinkPad-T470p:215359] [16] MLMC-M++(+0x3e979)[0x55f643e37979]
[niklas-ThinkPad-T470p:215359] [17] MLMC-M++(+0x3ebca)[0x55f643e37bca]
[niklas-ThinkPad-T470p:215359] [18] MLMC-M++(+0x369a7)[0x55f643e2f9a7]
[niklas-ThinkPad-T470p:215359] [19] /lib/x86_64-linux-gnu/libc.so.6(__libc_start_main+0xf3)[0x7f80bbe080b3]
[niklas-ThinkPad-T470p:215359] [20] MLMC-M++(+0x34c6e)[0x55f643e2dc6e]
[niklas-ThinkPad-T470p:215359] *** End of error message ***
--------------------------------------------------------------------------
Primary job terminated normally, but 1 process returned
a non-zero exit code. Per user-direction, the job has been aborted.
--------------------------------------------------------------------------
--------------------------------------------------------------------------
mpirun noticed that process rank 0 with PID 0 on node niklas-ThinkPad-T470p exited on signal 11 (Segmentation fault).
--------------------------------------------------------------------------
---------------------------------------------------------------------------
KeyError Traceback (most recent call last)
<ipython-input-14-45edc9a032ef> in <module>()
9 mpp.parse_log()
10
---> 11 convergence_plot(mpp)
~/CLionProjects/mlmc/python/mlmc_mppy.py in convergence_plot(mpp, label, save_only)
137 raise ValueError('Label not found in mpp data')
138 if label is None:
--> 139 mpp.data['Parsed'] = range(1, len(mpp.data['Used Levels']) + 1)
140 label = 'Parsed'
141
KeyError: 'Used Levels'
%% Cell type:markdown id: tags:
### Dependency on $\sigma$
%% Cell type:code id: tags:
``` python
mpp.clean_data()
mpp.reset_data()
sigmas = [1.0, 1.5, 2.0]
for sigma in sigmas:
kwargs["sigma"] = sigma
mpp.run(config="mlmc_elliptic", kwargs=kwargs)
mpp.parse_log()
convergence_plot(mpp)
```
%% Cell type:markdown id: tags:
### Convergence tests - dependency on $\lambda$
%% Cell type:code id: tags:
``` python
mpp.clean_data()
mpp.reset_data()
lambdas = [0.05, 0.10, 0.15]
for lamb in lambdas:
kwargs["lambda"] = [lamb, lamb]
mpp.run(config="mlmc_elliptic", kwargs=kwargs)
mpp.parse_log()
convergence_plot(mpp)
```
%% Cell type:markdown id: tags:
### Multilevel Monte Carlo - Experiments with $s = 1.4$
%% Cell type:code id: tags:
``` python
mpp.clean_data()
mpp.reset_data()
kwargs={"smoothing": 1.4,
"maxLevel": 10}
for epsilon in epsilons:
kwargs["epsilon"] = epsilon
mpp.run(config="mlmc_elliptic", kwargs=kwargs)
mpp.parse_log()
convergence_plot(mpp)
mlmc_plot(mpp)
```
%% Cell type:markdown id: tags:
### Multilevel Monte Carlo - Experiments with $s = 1.8$
%% Cell type:code id: tags:
``` python
mpp.clean_data()
mpp.reset_data()
kwargs={"smoothing": 1.8,
"maxLevel": 10}
for epsilon in epsilons:
kwargs["epsilon"] = epsilon
mpp.run(config="mlmc_elliptic", kwargs=kwargs)
mpp.parse_log()
convergence_plot(mpp)
mlmc_plot(mpp)
```
%% Cell type:markdown id: tags:
# Results from paper as reference
%% Cell type:markdown id: tags:
<img src="../tex/img/img.png"
alt="Convergence Results Paper"
width = 50%
style="float: center;" />
%% Cell type:markdown id: tags:
<img src="../tex/img/img2.png"
alt="MLMC Results Paper"
width = 50%
style="float: center;" />
......
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